The defender reportedly left the Etihad Stadium on crutches after suffering a knee injury in the second half of City's Champions League draw with Real Madrid.
The 26-year-old briefly left the field for treatment 10 minutes from time but played on until the end of the match.
Kompany received further treatment after the final whistle and was unable to conduct post-match interviews with the press.
But he did later relay his thoughts on the match, which saw City's hopes of progress in the competition end.
City had needed to beat the Spanish giants to stand any chance of reaching the knockout stages but could only manage a 1-1 draw.
It is the second successive season the club have failed to make it beyond the group phase.
Kompany wrote on Twitter: "The opportunity to claim our place at the top of European football will come again. No sad faces, just the hunger to improve."